Harmonic Analysis on Symmetric Spaces-Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

This book offers an introduction to harmonic analysis on the simplest symmetric spaces. It places an emphasis on motivation, concrete examples, history, and, above all, applications in mathematics, statistics, physics, and engineering.

This unique text is an introduction to harmonic analysis on the simplest symmetric spaces, namely Euclidean space, the sphere, and the Poincaré upper half plane. This book is intended for beginning graduate students in mathematics or researchers in physics or engineering. Written with an informal style, the book places an emphasis on motivation, concrete examples, history, and, above all, applications in mathematics, statistics, physics, and engineering.

Many corrections and updates have been incorporated in this new edition. Updates include discussions of P. Sarnak and others' work on quantum chaos, the work of T. Sunada, Marie-France Vignéras, Carolyn Gordon, and others on Mark Kac's question "Can you hear the shape of a drum?", A. Lubotzky, R. Phillips and P. Sarnak's examples of Ramanujan graphs, and, finally, the author's comparisons of continuous theory with the finite analogues.

Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, Poisson's summation formula and applications in crystallography and number theory, applications of spherical harmonic analysis to the hydrogen atom, the Radon transform, non-Euclidean geometry on the Poincaré upper half plane H or unit disc and applications to microwave engineering, fundamental domains in H for discrete groups , tessellations of H from such discrete group actions, automorphic forms, and the Selberg trace formula and its applications in spectral theory as well as number theory.

New edition extensively revised and updated Features many new examples and figures (such as density plots of modular forms as well as finite analogues) New topics include: wavelets, quasicrystals, Ramanujan graphs, elliptic curves, modular knots, triangle and quaternion groups, comparisons of the continuous theory with its finite analogues, and recent progress on Mark Kac's question "Can you hear the shape of a drum?" Includes supplementary material: sn.pub/extras

Autorentext
Audrey Anne Terras is currently Professor Emerita of Mathematics at the University of California at San Diego.

Inhalt
Chapter 1 Flat Space. Fourier Analysis on R^m..- 1.1 Distributions or Generalized Functions.- 1.2 Fourier Integrals.- 1.3 Fourier Series and the Poisson Summation Formula.- 1.4 Mellin Transforms, Epstein and Dedekind Zeta Functions.- 1.5 Finite Symmetric Spaces, Wavelets, Quasicrystals, Weyls Criterion for Uniform Distribution.- Chapter 2 A Compact Symmetric Space--The Sphere.- 2.1 Fourier Analysis on the Sphere.- 2.2 O(3) and R^3. The Radon Transform.- Chapter 3 The Poincaré Upper Half-Plane.- 3.1 Hyperbolic Geometry.- 3.2 Harmonic Analysis on H.- 3.3 Fundamental Domains for Discrete Subgroups of G = SL(2, R).- 3.4 Modular of Automorphic Forms--Classical.- 3.5 Automorphic Forms--Not So Classical--Maass Waveforms.- 3.6 Modular Forms and Dirichlet Series. Hecke Theory and Generalizations.- References.- Index.